Symplectic homology and Stein manifolds

辛同调和斯坦因流形

• 批准号: 1005365
• 负责人: Mark McLean
• 金额: $ 13.01万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2013-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1005365&HistoricalAwards=false
• 关键词: Symplectic; homology; Stein; manifolds

AbstractAward: DMS-1005365Principal Investigator: Mark McLeanThe subject area of this project is the symplectic geometry of Stein manifolds. A Stein manifold is a properly embedded complex submanifold of complex affine space. This has a symplectic form induced from the standard one in affine space. If we take a large sphere and intersect it with this Stein manifold then we get another manifold called a Stein fillable contact manifold. Important progress in studying Stein manifolds symplectically was achieved by Eliashberg and Gromov. The primary aim of this project is to find exotic Stein structures and Stein fillable contact structures. The PI intends to prove that there are uncountably many symplectically different Stein structures diffeomorphic to an even dimensional manifold admitting a proper and bounded from below Morse function with only finitely many critical points of index at most half its dimension. The PI also intends to prove that there are infinitely many Stein fillable contact structures on each odd dimensional sphere of dimension 5 and higher, and more generally on Stein fillable contact manifolds obtained from affine varieties. The PI will use an invariant of Stein manifolds called symplectic homology to distinguish these. The PI also aims to show that there is no algorithm to tell you whether one Stein manifold diffeomorphic to affine space is symplectomorphic to another one diffeomorphic to affine space of complex dimension greater than 6. The PI also aims prove a similar undecidability result for contact structures on all odd dimensional spheres of dimension greater than 13. The PI will use an invariant called the growth rate of symplectic homology to achieve this. The PI will use growth rates to show that certain cotangent bundles have many Reeb orbits (even degenerate ones). This generalizes the Gromoll-Meyer theorem. The PI will show that the cotangent bundle of a rationally hyperbolic manifold is not symplectomorphic to a smooth affine variety using growth rates.If we have some classical system such as a pendulum then at any point in time it has a particular position and momentum. If this system has many moving parts such as a double pendulum or a collection free particles then it has many positions and momenta. The set of all such positions and momenta can be encoded in an object called a symplectic manifold. For example the symplectic manifold associated to a pendulum turns out to be a cylinder. Symplectic manifolds are important in many areas of physics such as quantum mechanics and String theory. The PI will study a large class of symplectic manifolds obtained from objects called Stein manifolds. The PI will construct a large list of Stein manifolds called exotic Stein manifolds which look very similar to the symplectic manifold coming from a set of free particles but are actually different if we look at the motion of their respective classical systems. The PI intends to show that there is no computer algorithm telling you if two given exotic Stein manifolds come from the same classical system. This result is useful because it tells us that certain classical systems are very hard to study in general.

摘要奖：DMS-1005365首席研究员：马克麦克莱恩这个项目的主题领域是辛几何的斯坦流形。Stein流形是复仿射空间的适当嵌入的复子流形。这有一个辛形式从标准的仿射空间。如果我们取一个大的球面，和这个Stein流形相交，那么我们得到另一个流形，叫做Stein可填充接触流形。Eliashberg和Gromov在辛地研究Stein流形方面取得了重要进展。本项目的主要目的是寻找奇异的Stein结构和Stein可填充接触结构。 PI的目的是证明，有无穷多个辛不同的Stein结构同构于一个偶数维流形，承认一个适当的和有界的从下面的莫尔斯函数，只有100多个临界点的指数最多一半的维度。PI还打算证明，有无穷多个斯坦可填充接触结构上的每个奇数维球体的尺寸5和更高，更一般的斯坦可填充接触流形从仿射品种。PI将使用称为辛同调的Stein流形不变量来区分这些。PI还旨在表明，没有算法可以告诉你一个仿射空间的Stein流形是否与另一个复维数大于6的仿射空间的Stein流形辛同构。PI的目的也是为了证明一个类似的不可判定性的结果，接触结构的所有奇数维球体的尺寸大于13。PI将使用称为辛同调增长率的不变量来实现这一点。PI将使用增长率来证明某些余切丛有许多Reeb轨道（甚至退化的）。这推广了Gromoll-Meyer定理。PI将表明，一个有理双曲流形的余切丛不是辛形的光滑仿射簇使用的增长率。如果我们有一些经典系统，如钟摆，那么在任何时间点，它有一个特定的位置和动量。如果这个系统有许多运动部件，比如一个双摆或一个自由粒子的集合，那么它就有许多位置和动量。所有这些位置和动量的集合可以被编码在一个称为辛流形的对象中。例如，与钟摆相关联的辛流形原来是圆柱体。辛流形在物理学的许多领域都很重要，例如量子力学和弦理论。PI将研究从称为Stein流形的对象获得的一大类辛流形。PI将构建一个称为奇异Stein流形的Stein流形的大列表，这些流形看起来非常类似于来自一组自由粒子的辛流形，但如果我们观察它们各自经典系统的运动，它们实际上是不同的。PI旨在表明，没有计算机算法告诉你，如果两个给定的奇异Stein流形来自同一个经典系统。这个结果是有用的，因为它告诉我们，某些经典系统是很难研究的一般。